Over the years I've converted to the belief that what is important in language and grammar is that the communication is clearly understood and not unintentionally ambiguous, not that it satisfies any formal criterion. Whether you say your mum or my mum, no one is going to be confused by what you mean. So use whichever feels right to you.
The subtle distinction is if Bob knows Andy's mum and, for example, Andy's mum is the kind to hold a grudge and get revenge while Bob's mum is unusually forgiving and generous, well, Bob might be trying to convey that if Andy had said bad things to Bob's mum, he is better off forgetting about it, but having said bad things to Andy's mum, he better make amends to "your mum". If Bob does not know Andy's mum, and it is Bob's mum that is vengeful, he should probably say "my mum", but I would not lead in with "if I were you", I would instead say "If I had said bad things to my mum, I'd apologize quickly."
Andy: I just spent $5 on the worst cup of coffee ever.
Bob: If I were you, I'd want my money back.
Seems totally wrong (or maybe just British) to me to say If I were you, I'd want your money back, but formally it is completely analogous to saying If I were you, I'd apologize to your mum.
On the other hand, I agree with keshlam that it is harder to get confused when you say "your mum" because if Bob were Andy, who's mother would be "your mum"? There's no "you" left in that situation, so there is only "my mom". "Your mum" must still refer to Andy's mum.
Also, saying "your mum" feels more empathetic and keeps the focus on Andy's mum rather than discussing how Bob would treat Bob's mum, so I would probably say "your mum" even though I would never say "your money". I (Bob) would say "my mum" only when referring my my (Bob's) mum, but if I meant that, meaning I was talking about how I would treat my mum if I had said bad things to her, I would not say "if I were you", I would say "if it were me..." or "if I were in your situation, I'd apologize to my mum". If I were you, I wouldn't have my mum, I'd have your mum. :-)
Back to the original dialog, "my mum" could mean either Andy's mum or Bob's mum, but in either case the message is the same: If I had said bad things to my mum, I would apologize to my mum. The idea that Andy should apologize to Bob's mum is so absurd that the statement If I were you, I'd apologize to my mum is not ambiguous in practice. No one would ever think that Bob is suggesting Andy get Bob's money back for the coffee, right?
Another way to think about it is what if the lead in had been:
Andy: I've said bad things to your mum.
Bob: If I were you, I'd apologise to my mum.
Again, it would be nonsense to think Bob is suggesting Andy apologize to Andy's mom. "My" has to continue to refer to "Bob". You have to look at the context of the sentence to understand it either way.
In the end, either way the message is clear. That's all that really matters.